From Theorem  REF , we see the main thing we have to do is to find a appropriate upper bounds and lower bounds of \(u_n\) .
We will use the asymptotic estimation of P-recursive sequences given by Birkhoff and Trjitzinsky [1]} and developed by Wimp and Zeilberger [2]}. They showed that a P-recursive sequence is asymptotically equal to a linear combination of terms of the form
\( e^{Q(\rho ,n)} s(\rho ,n),\) 
